APPLICABLE TO THE MOTION OF FLUIDS. 387 



area, such in magnitude and position that the straight lines joining the 

 corresponding angular points of the two areas are normals to the first 

 surface. By the nature of curve surfaces these normals will meet two 

 and two in two focal lines situated in the planes of greatest and least 

 curvature, and cutting the normals at right angles. Let the small area 

 of which P is the centre be m?, and let r, r be the distances of the focal 

 lines from P. Then if the positions of the focal lines do not vary with 



the time, the other area is ultimately — £ > the interval 



between the two surfaces of displacement being a given small quantity 

 ir. This is the case of rectilinear motion. But if the direction of the 

 motion through P is continually changing, the surface of displacement 

 through that point will vary with the time. Hence the positions of 

 the focal lines and the magnitudes of r and r' will change continually, 

 whilst the area m* may be supposed to remain the same and always 

 to pass through the point P. Let r and r' represent the values of the 

 principal radii of curvature at the time t, and let a and fr be the velo- 

 cities of the focal lines estimated in the direction of the radii of curva- 

 ture and considered positive when the motion is towards P. Then at 

 the time t + it the values of r and r become r — ait and r' — frit, 

 and the elementary area on the second surface is 



, (r + ir - ait) {r' + ir- frit) 

 m - {r - ait) (r' - frit) 



which is equal to 



/ _ ait N / _ _fri±_\ 

 , (r + ir) (r' + ir) V r + irl \ r + ir ) 



- "■ ? * Mw-£) ' 



or , „.fc£Mpir) . (l + ^ (l + 2i^) ultimateIy . 



Hence, by omitting quantities of the order of x — , the result is 



the same as when the position of the focal lines is supposed to be fixed. 

 If therefore V and p be the velocity and density of the fluid which 



